Finite dimensional Hopf actions on algebraic quantizations
Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra, we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z 1 ,… z s ] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s = 0. We a...
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| Format: | Article |
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Mathematical Sciences Publishers
2018
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| Online Access: | http://hdl.handle.net/1721.1/115879 https://orcid.org/0000-0002-0710-1416 |
| _version_ | 1826189253934055424 |
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| author | Etingof, Pavel I Walton, Chelsea |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I Walton, Chelsea |
| author_sort | Etingof, Pavel I |
| collection | MIT |
| description | Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra, we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z 1 ,… z s ] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s = 0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal enveloping algebras of finite dimensional Lie algebras, spherical symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras (in particular, quantized quiver varieties), finite W-algebras and their central reductions, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, and Sklyanin algebras. The generalization in the last three cases uses a result from algebraic number theory due to A. Perucca. |
| first_indexed | 2024-09-23T08:11:54Z |
| format | Article |
| id | mit-1721.1/115879 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T08:11:54Z |
| publishDate | 2018 |
| publisher | Mathematical Sciences Publishers |
| record_format | dspace |
| spelling | mit-1721.1/1158792022-09-23T11:34:04Z Finite dimensional Hopf actions on algebraic quantizations Etingof, Pavel I Walton, Chelsea Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I Walton, Chelsea Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra, we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z 1 ,… z s ] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s = 0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal enveloping algebras of finite dimensional Lie algebras, spherical symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras (in particular, quantized quiver varieties), finite W-algebras and their central reductions, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, and Sklyanin algebras. The generalization in the last three cases uses a result from algebraic number theory due to A. Perucca. National Science Foundation (U.S.) (Grant CHE-1464804) National Science Foundation (U.S.) (Grant DMS-1550306) 2018-05-24T20:16:19Z 2018-05-24T20:16:19Z 2016-12 2016-05 2018-05-21T14:51:31Z Article http://purl.org/eprint/type/JournalArticle 1944-7833 1937-0652 http://hdl.handle.net/1721.1/115879 Etingof, Pavel and Chelsea Walton. “Finite Dimensional Hopf Actions on Algebraic Quantizations.” Algebra & Number Theory 10, 10 (December 2016): 2287–2310 © 2016 Mathematical Sciences Publishers https://orcid.org/0000-0002-0710-1416 http://dx.doi.org/10.2140/ANT.2016.10.2287 Algebra & Number Theory Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Mathematical Sciences Publishers arXiv |
| spellingShingle | Etingof, Pavel I Walton, Chelsea Finite dimensional Hopf actions on algebraic quantizations |
| title | Finite dimensional Hopf actions on algebraic quantizations |
| title_full | Finite dimensional Hopf actions on algebraic quantizations |
| title_fullStr | Finite dimensional Hopf actions on algebraic quantizations |
| title_full_unstemmed | Finite dimensional Hopf actions on algebraic quantizations |
| title_short | Finite dimensional Hopf actions on algebraic quantizations |
| title_sort | finite dimensional hopf actions on algebraic quantizations |
| url | http://hdl.handle.net/1721.1/115879 https://orcid.org/0000-0002-0710-1416 |
| work_keys_str_mv | AT etingofpaveli finitedimensionalhopfactionsonalgebraicquantizations AT waltonchelsea finitedimensionalhopfactionsonalgebraicquantizations |